STAT 240 PROBLEM SOLVING SESSION #2

Conditional Probability

Example: Statistical Independence Suppose that women obtain 54% of all bachelor’s degrees in a particular country and that 20% of all bachelor’s degrees are in engineering. Also, 8% of all bachelor’s degrees go to women majoring in engineering. Are the events “the bachelor’s degree holder is a woman” and “the bachelor’s degree is in engineering” statistically independent?

Conditional Probability The question, "Do you smoke?" was asked of 100 people. Results are shown in the table. Yes NoTotal Male Female Total

Example : Conditional Probability You are off to soccer, and want to be the Goalkeeper, but that depends who is the Coach today:  with Coach Sam the probability of being Goalkeeper is 0,5  with Coach Alex the probability of being Goalkeeper is 0,3  Sam is Coach more often... about 6 out of every 10 games. So, what is the probability you will be a Goalkeeper today?

Example : Conditional Probability Let's build a tree diagram. First we show the two possible coaches: Sam Alex

Example : Conditional Probability Now, if you get Sam, there is 0,5 probability of being Goalie: Sam Yes No ??

Example : Conditional Probability If you get Alex, there is 0.3 probability of being Goalie: Sam Alex No Yes ?? No Yes 0.6 X 0.5 =0.3

Example : Conditional Probability If you get Alex, there is 0.3 probability of being Goalie: Sam Alex No Yes 0.4 X 0.3 = 0.12 No Yes 0.6 X 0.5 =0.3

Example : Conditional Probability If you get Alex, there is 0.3 probability of being Goalie: Sam Alex No Yes 0.4 X 0.3 = 0.12 No Yes 0.6 X 0.5 = = 0.42

Example : Conditional Probability

Conditioning and Bayes' Theorem A

Example : Bayes' Theorem Three jars contain colored balls as described in the table below. One jar is chosen at random and a ball is selected. If the ball is red, what is the probability that it came from the 2 nd jar?

Example : Bayes' Theorem We will define the following: J 1 is the event that first jar is chosen J 2 is the event that second jar is chosen J 3 is the event that third jar is chosen R is the event that a red ball is selected

Example : Bayes' Theorem Let’s look at the Venn Diagram

Example : Bayes' Theorem All of the red balls are in the first, second, and third jar so their set overlaps all three sets of our partition

Example : Bayes' Theorem

Updating our Venn Diagram with these probabilities:

Example : Bayes' Theorem

Example: QUIZ QUESTION Consider a medical test for a disease. The test has a probability of 0.95 of currently (or positively) detecting an infected person (sensitivity). It has a probability of 0.90 of identifying a healthy person (specificity). In the population 3% have the disease. a)What is the probability that a person testing positive is actually infected. b)What is the probability that a person testing negative is actually infected.